5.3 The exponential distribution  (Page 6/25)

Chapter review

If X has an exponential distribution with mean μ , then the decay parameter is m = $\frac{1}{\mu }$ , and we write X ∼ Exp ( m ) where x ≥ 0 and m >0 . The probability density function of X is f ( x ) = me ^-mx (or equivalently $f(x)=\frac{1}{\mu }{e}^{-x/\mu }$ . The cumulative distribution function of X is P ( X ≤ x ) = 1 – e ^{– mx} .

The exponential distribution has the memoryless property , which says that future probabilities do not depend on any past information. Mathematically, it says that P ( X > x + k | X > x ) = P ( X > k ).

If T represents the waiting time between events, and if T ∼ Exp ( λ ), then the number of events X per unit time follows the Poisson distribution with mean λ . The probability density function of P X is $(X=k)=\frac{{\lambda }^k{e}^{-k}}{k!}$ . This may be computed using a TI-83, 83+, 84, 84+ calculator with the command poissonpdf( λ , k ). The cumulative distribution function P ( X ≤ k ) may be computed using the TI-83, 83+,84, 84+ calculator with the command poissoncdf( λ , k ).

Formula review

Exponential: X ~ Exp ( m ) where m = the decay parameter

• pdf: f ( x ) = me ^{(– mx )} where x ≥ 0 and m >0
• cdf: P ( X ≤ x ) = 1 – e ^{(– mx )}
• mean µ = $\frac{1}{m}$
• standard deviation σ = µ
• percentile k : k = $\frac{ln(1-AreaToTheLeftOfk)}{(-m)}$
• Additionally
  • P ( X > x ) = e ^{(– mx )}
  • P ( a < X < b ) = e ^{(– ma )} – e ^{(– mb )}
• Memoryless Property: P ( X > x + k | X > x ) = P ( X > k )
• Poisson probability: $P(X=k)=\frac{{\lambda }^k{e}^{-k}}{k!}$ with mean λ
• k ! = k *( k -1)*( k -2)*( k -3)…3*2*1

References

Data from the United States Census Bureau.

Data from World Earthquakes, 2013. Available online at http://www.world-earthquakes.com/ (accessed June 11, 2013).

“No-hitter.” Baseball-Reference.com, 2013. Available online at http://www.baseball-reference.com/bullpen/No-hitter (accessed June 11, 2013).

Zhou, Rick. “Exponential Distribution lecture slides.” Available online at www.public.iastate.edu/~riczw/stat330s11/lecture/lec13.pdf‎ (accessed June 11, 2013).

Use the following information to answer the next ten exercises. A customer service representative must spend different amounts of time with each customer to resolve various concerns. The amount of time spent with each customer can be modeled by the following distribution: X ~ Exp (0.2)

Use the following information to answer the next seven exercises. A distribution is given as X ~ Exp (0.75).

Draw the distribution.

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Use the following information to answer the next 16 exercises. Carbon-14 is a radioactive element with a half-life of about 5,730 years. Carbon-14 is said to decay exponentially. The decay rate is 0.000121. We start with one gram of carbon-14. We are interested in the time (years) it takes to decay carbon-14.